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Theorem elab6g 3652
Description: Membership in a class abstraction. Class version of sb6 2080. (Contributed by SN, 5-Oct-2024.)
Assertion
Ref Expression
elab6g (𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem elab6g
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2813 . 2 (𝑦 = 𝐴 → (𝑦 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜑}))
2 eqeq2 2736 . . . 4 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
32imbi1d 341 . . 3 (𝑦 = 𝐴 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝐴𝜑)))
43albidv 1915 . 2 (𝑦 = 𝐴 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
5 df-clab 2702 . . 3 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
6 sb6 2080 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
75, 6bitri 275 . 2 (𝑦 ∈ {𝑥𝜑} ↔ ∀𝑥(𝑥 = 𝑦𝜑))
81, 4, 7vtoclbg 3537 1 (𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531   = wceq 1533  [wsb 2059  wcel 2098  {cab 2701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802
This theorem is referenced by:  elabd2  3653  elabgt  3655  elabg  3659  sbc6g  3800  upwordnul  46140  upwordsing  46144
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